Here we have explicitated \(\lambda = \frac{i m a_0^2 \eta ^2}{2}\) . The functions \(\phi (a, b, z)\)  and \(\psi (a,b,z)\)  are the confluent hypergeometric functions (also denoted respectively as \(M(a,b,z)\)  and \(U(a,b,z)\)  [1]}). At early times, we impose the normalization condition and we assume that the modes are positive with respect to \(\partial _{\eta }\) . We obtain
\(\chi _{k}=\frac{(2\pi )^{\frac{3}{2}}}{\sqrt{4ma^{2}_{0}\eta }}e^{-\frac{\pi k^2}{4ma^{2}_{0}}}\bigg (\frac{ima^{2}_{0}\eta ^{2}}{2}\bigg )^{\frac{1}{4}}e^{\frac{-ima^{2}_{0}\eta ^{2}}{4}}\psi \left(\frac{1}{4}\left(\frac{-2k^2}{ima^{2}_{0}}+1\right),\frac{1}{2},\frac{ima^{2}_{0}\eta ^{2}}{2}\right)\,.\) 
